House Majority Leader Eric Cantor (R-Va.) didn’t comment on the Senate principles that include a pathway to citizenship for illegal immigrants — he’ll lay out some of his own next week at the American Enterprise Institute.

Rep. Bob Goodlatte, the Virginia Republican who chairs the House Judiciary Committee tasked with immigration policy, said “we have a lot to discuss” when considering proposals “ that deal with the legal status of 11 million illegal immigrants currently living in the U.S.” His panel will explore the nation’s immigration system, he said.

Some House Republicans have come around to supporting “amnesty” or a pathway to citizenship for the millions of illegal immigrants already in the country who could seek legal residency under the Senate plan. But there are still hardline conservatives who oppose it and will undoubtedly make the Senate plan a tough sell in the House. And the Senate plan could fall apart before it even gets there.

Rep. Raul Labrador, a Hispanic Republican from Idaho who has been working on immigration policy for his party, illustrates the uphill battle faced by the Senate deal — which would allow undocumented immigrants to register for permanent legal residency after certain border enforcement requirements were met — in the House.

In an interview Monday, he said he supports the principles that the Senate laid out, but flatly said “creating a new pathway” to citizenship for undocumented workers “is not a good idea.” It would encourage more illegal immigration, he said.

Asked if he was flexible, Labrador said: “The question that is more appropriate is how flexible are they? We’ve gone a bit to their side. If they’re unwilling to be flexible on that issue, they want political victory not policy victory.”

Those words aren’t exactly ringing endorsements from key Republican leaders at a moment when the political stars seem to be aligning in favor of reform.

House Republicans realize they have political and substantive reason to embrace a deal on the hot-button issue after losing Latinos by record numbers in 2012, helping reelect Barack Obama and keep them out of the Senate majority. But top House Republicans favor a more incremental approach to legislating the issue, differ on the specifics of key policies and are bound to respect the arduous committee process that favors deliberation instead of backroom deals.

The Senate’s proposal would seek to overhaul the legal immigration system as well as create a pathway to citizenship for the nation’s roughly 11 million illegal immigrants. But establishing that pathway would depend on whether the U.S. first implements stricter border enforcement measures and new rules ensuring immigrants have left the country if they’ve overstayed their visas. Young people brought to the country as children illegally and seasonal agriculture industry workers would have a faster path to citizenship.

“I am confident the majority in both houses, led by the president of the United States, who made this a major campaign issue, that we will succeed,” said John McCain (R-Ariz.) on Monday, who’s a member of the Senate group. “But we’re not going to get everybody on board.”